A358912 Number of finite sequences of integer partitions with total sum n and all distinct lengths.

1, 1, 2, 5, 11, 23, 49, 103, 214, 434, 874, 1738, 3443, 6765, 13193, 25512, 48957, 93267, 176595, 332550, 622957, 1161230, 2153710, 3974809, 7299707, 13343290, 24280924, 43999100, 79412942, 142792535, 255826836, 456735456, 812627069, 1440971069, 2546729830

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Example

The a(1) = 1 through a(4) = 11 sequences:
  (1)  (2)   (3)      (4)
       (11)  (21)     (22)
             (111)    (31)
             (1)(11)  (211)
             (11)(1)  (1111)
                      (11)(2)
                      (1)(21)
                      (2)(11)
                      (21)(1)
                      (1)(111)
                      (111)(1)

Mathematica

ptnseq[n_]:=Join@@Table[Tuples[IntegerPartitions/@comp], {comp, Join@@Permutations/@IntegerPartitions[n]}];
Table[Length[Select[ptnseq[n], UnsameQ@@Length/@#&]], {n, 0, 10}]

Prog

(PARI)
P(n, y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n, y)); [subst(serlaplace(p), y, 1) | p<-Vec(prod(k=1, n, 1 + y*polcoef(g, k, y) + O(x*x^n)))]} \\ Andrew Howroyd, Dec 30 2022

Crossrefs

The case of set partitions is A007837.

This is the case of A055887 with all distinct lengths.

For distinct sums instead of lengths we have A336342.

The case of twice-partitions is A358830.

The unordered version is A358836.

The version for constant instead of distinct lengths is A358905.

A000041 counts integer partitions, strict A000009.

A063834 counts twice-partitions.

A141199 counts sequences of partitions with weakly decreasing lengths.

Cf. A000219, A001970, A038041, A060642, A218482, A271619, A319066, A358831, A358901, A358906, A358908.

Sequence in context: A034468 A283075 A130668 * A083380 A018112 A274489

Adjacent sequences: A358909 A358910 A358911 * A358913 A358914 A358915

Keyword

nonn

Author

Gus Wiseman, Dec 07 2022

Extensions

Terms a(16) and beyond from Andrew Howroyd, Dec 30 2022

Status

approved